Optimal. Leaf size=134 \[ -\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{3/4}}+\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{\sqrt {a} c^{3/4}} \]
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Rubi [A]
time = 0.07, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {714, 1144, 214}
\begin {gather*} \frac {\sqrt {\sqrt {a} e+\sqrt {c} d} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{\sqrt {a} c^{3/4}}-\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 714
Rule 1144
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x}}{a-c x^2} \, dx &=(2 e) \text {Subst}\left (\int \frac {x^2}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt {d+e x}\right )\\ &=-\left (\left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) \text {Subst}\left (\int \frac {1}{c d-\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )\right )+\left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \text {Subst}\left (\int \frac {1}{c d+\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )\\ &=-\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{3/4}}+\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{\sqrt {a} c^{3/4}}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 156, normalized size = 1.16 \begin {gather*} \frac {-\sqrt {-c d-\sqrt {a} \sqrt {c} e} \tan ^{-1}\left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )+\sqrt {-c d+\sqrt {a} \sqrt {c} e} \tan ^{-1}\left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {a} c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.50, size = 143, normalized size = 1.07
method | result | size |
derivativedivides | \(-2 e c \left (-\frac {\left (c d +\sqrt {a c \,e^{2}}\right ) \arctanh \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-c d +\sqrt {a c \,e^{2}}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\) | \(143\) |
default | \(-2 e c \left (-\frac {\left (c d +\sqrt {a c \,e^{2}}\right ) \arctanh \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-c d +\sqrt {a c \,e^{2}}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\) | \(143\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 339 vs.
\(2 (100) = 200\).
time = 1.03, size = 339, normalized size = 2.53 \begin {gather*} \frac {1}{2} \, \sqrt {\frac {a c \sqrt {\frac {1}{a c^{3}}} e + d}{a c}} \log \left (a c^{2} \sqrt {\frac {a c \sqrt {\frac {1}{a c^{3}}} e + d}{a c}} \sqrt {\frac {1}{a c^{3}}} e + \sqrt {x e + d} e\right ) - \frac {1}{2} \, \sqrt {\frac {a c \sqrt {\frac {1}{a c^{3}}} e + d}{a c}} \log \left (-a c^{2} \sqrt {\frac {a c \sqrt {\frac {1}{a c^{3}}} e + d}{a c}} \sqrt {\frac {1}{a c^{3}}} e + \sqrt {x e + d} e\right ) - \frac {1}{2} \, \sqrt {-\frac {a c \sqrt {\frac {1}{a c^{3}}} e - d}{a c}} \log \left (a c^{2} \sqrt {-\frac {a c \sqrt {\frac {1}{a c^{3}}} e - d}{a c}} \sqrt {\frac {1}{a c^{3}}} e + \sqrt {x e + d} e\right ) + \frac {1}{2} \, \sqrt {-\frac {a c \sqrt {\frac {1}{a c^{3}}} e - d}{a c}} \log \left (-a c^{2} \sqrt {-\frac {a c \sqrt {\frac {1}{a c^{3}}} e - d}{a c}} \sqrt {\frac {1}{a c^{3}}} e + \sqrt {x e + d} e\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.08, size = 76, normalized size = 0.57 \begin {gather*} - 2 e \operatorname {RootSum} {\left (256 t^{4} a^{2} c^{3} e^{4} - 32 t^{2} a c^{2} d e^{2} - a e^{2} + c d^{2}, \left ( t \mapsto t \log {\left (- 64 t^{3} a c^{2} e^{2} + 4 t c d + \sqrt {d + e x} \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.51, size = 153, normalized size = 1.14 \begin {gather*} \frac {\sqrt {-c^{2} d - \sqrt {a c} c e} {\left | c \right |} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {c d + \sqrt {c^{2} d^{2} - {\left (c d^{2} - a e^{2}\right )} c}}{c}}}\right )}{\sqrt {a c} c^{2}} - \frac {\sqrt {-c^{2} d + \sqrt {a c} c e} {\left | c \right |} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {c d - \sqrt {c^{2} d^{2} - {\left (c d^{2} - a e^{2}\right )} c}}{c}}}\right )}{\sqrt {a c} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.34, size = 302, normalized size = 2.25 \begin {gather*} -2\,\mathrm {atanh}\left (\frac {2\,\left (\left (16\,c^3\,d^2\,e^2+16\,a\,c^2\,e^4\right )\,\sqrt {d+e\,x}-\frac {16\,c\,d\,e^2\,\left (e\,\sqrt {a^3\,c^3}+a\,c^2\,d\right )\,\sqrt {d+e\,x}}{a}\right )\,\sqrt {\frac {e\,\sqrt {a^3\,c^3}+a\,c^2\,d}{4\,a^2\,c^3}}}{16\,c^2\,d^2\,e^3-16\,a\,c\,e^5}\right )\,\sqrt {\frac {e\,\sqrt {a^3\,c^3}+a\,c^2\,d}{4\,a^2\,c^3}}-2\,\mathrm {atanh}\left (\frac {2\,\left (\left (16\,c^3\,d^2\,e^2+16\,a\,c^2\,e^4\right )\,\sqrt {d+e\,x}+\frac {16\,c\,d\,e^2\,\left (e\,\sqrt {a^3\,c^3}-a\,c^2\,d\right )\,\sqrt {d+e\,x}}{a}\right )\,\sqrt {-\frac {e\,\sqrt {a^3\,c^3}-a\,c^2\,d}{4\,a^2\,c^3}}}{16\,c^2\,d^2\,e^3-16\,a\,c\,e^5}\right )\,\sqrt {-\frac {e\,\sqrt {a^3\,c^3}-a\,c^2\,d}{4\,a^2\,c^3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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